Abstract
Let be a Banach space. Let and denote by the Banach space of all -valued Bochner -integrable functions on a certain positive complete -finite measure space , endowed with the usual -norm. In this paper, the theory of lifting is used to prove that, for any weakly compact subset of , the set is -simultaneously proximinal in for any arbitrary monotonous norm in .
Dedicated to Hirak Chaabi Rifeño
1. Introduction
Throughout this paper, is a real Banach space and is the closed unit ball of , the dual of , with -topology. Let be a positive complete -finite measure space; is a lifting and is the associated lifting topology (a base for this topology is is a null set; see [1, p. 59]). For an and , will also be denoted by .
Let . We denote by the Banach space of all Bochner -integrable functions on with values in endowed with the usual norm for every (see [2]).
Let be a positive integer. We say that a norm in is monotonous if, for every , such that for ; we have
Let be a subset of ; we say that is a best –simultaneous approximation from of the vectors if for every . If every -tuple of vectors admits a best –simultaneous approximation from , then is said to be –simultaneously proximinal in . Of course, for , the preceding concepts are just best approximation and proximinality.
The problem of the best -simultaneous approximation in has been deeply and extensively studied; see, for example, [3–11]. When is a reflexive subspace of , it was proved in [3] that is –simultaneously proximinal in , where is the Lebesgue measure on . We have also obtained similar results, in the Banach space of -valued Bochner -integrable functions defined on , where is the restriction of to a certain sub--algebra of and is a reflexive space (see [8]).
Our purpose is to study the -simultaneous proximinality of the set defined by for certain subsets of .
2. Preliminaries
In this section we wish to include some facts which are fundamental in our study.
Definition 1. Let be vectors in ; one says that a sequence in is -simultaneously approximating to in , if
Lemma 2 ([8, Lemma ]). Let be vectors in , and let be a -simultaneously approximating sequence to in . Assume that is weakly convergent to . Then is a best -simultaneous approximation from of .
The next result gives a property of -simultaneously approximating sequences in the space . The following lemma is taken from [8, Lemma ], which was proved for the case when and .
Lemma 3 ([8, Lemma ]). Let be a subset of . Let be functions in , and let be an -simultaneously approximating sequence to in . If is a sequence of -measurable sets such that , then is also an -simultaneously approximating sequence to in .
We also need a result which is probably the best known “subsequence splitting lemma” in integrable function spaces.
Lemma 4 (Kadec-Pełczyński-Rosenthal, Lemma in [12]). If is a bounded sequence in , then there exists a subsequence of and a sequence of pairwise disjoint measurable sets such that is uniformly integrable.
3. Main Results
In this section we give the simple and direct proofs of the main results. These results are similar to [6]. However, in [6], more hypotheses are necessary and the proofs are totally different.
Theorem 5. Let be a weakly compact subset of . Then is –simultaneously proximinal in .
Proof. Let be functions in , and let be a -simultaneously approximating sequence to in . We have Since is a convergent sequence, by taking into account the fact that for each we have one deduces that is bounded in .
By Lemma 4, there exists a subsequence of and a sequence of pairwise disjoint measurable sets such that is uniformly integrable in .
On the one hand, so we have . Therefore, by Lemma 3, is a -simultaneously approximating sequence to in .
Let us denote for each .
Since is weakly compact, then the sequence for a.e. and has a weakly convergent subsequence, which again is denoted by . Let us denote by its weak limit for a.e. . Therefore for each the numerical function is -measurable. So is weakly -measurable. On the other hand, for each , is -essentially separably valued; that is, there exists with and such that is a norm separable subset of . For each let us pick a dense and countable subset, , of . Then the set is norm closed and separable. For every and we have . Since, is weak limit of for a.e. , we obtain that for a.e. . Thus is -essentially separably valued. Therefore, Pettis Measurability Theorem [2, p. 42] guarantees that the function is -measurable.
Since is weakly convergent to for a.e. , then is bounded and for a.e. . Using this result, the boundedness of , and Fatou’s lemma [13, Theorem , p. 131], we get .
We claim that is in the closed convex hull of in . Suppose this is not true. Then, by [14, Theorem , p. 65], there is such that Now, we will use the result that is the space of all bounded continuous mapping (see [1, Theorem , p. 94]); we get a.e. on . Using Egoroff’s theorem [13, Theorem , p. 110] and uniform integrability of , we get , a contradiction. By taking convex combinations of elements of , if necessary, we can assume that in . Therefore, by Lemma 2, we have the fact that is the best -simultaneous approximation from of .
Although the proof of the following theorem is similar to that of Theorem 5, we provide the proof here as a means for the reader to readily justify these assertions.
Theorem 6. Let be a weakly compact subset of . Then is –simultaneously proximinal in for each .
Proof. Let be functions in , and let be a -simultaneously approximating sequence to in . We have Notice that for each we have Since is a convergent sequence, we deduce that is bounded in . Therefore, is uniformly integrable.
Since is weakly compact, then the sequence for a.e. has a weakly convergent subsequence, which again is denoted by . Let us denote by its weak limit for a.e. . Therefore for each the numerical function is -measurable. So is weakly -measurable. On the other hand, for each , is -essentially separably valued; that is, there exists with and such that is a norm separable subset of . For each let us pick a dense and countable subset, , of . Then the set is norm closed and separable. For every and we have . Since is weak limit of for a.e. , we obtain that for a.e. . Thus is -essentially separably valued. Therefore, the Pettis Measurability Theorem [2, p. 42] guarantees that the function is -measurable.
Since is weakly convergent to for a.e. , then is bounded and for a.e. . Using this result, the boundedness of , and Fatou’s lemma [13, Theorem , p. 131], we get .
We claim that is in the closed convex hull of in . Suppose this is not true. Then, by [14, Theorem , p. 65], there is a such that Now, we will be using the result that is the space of all bounded continuous mapping (see [1, Theorem , p. 97]); we get a.e. on . Using Egoroff’s theorem [13, Theorem , p. 110] and uniform integrability of , we get , a contradiction. By taking convex combinations of elements of , if necessary, we can assume that in . Therefore, by Lemma 2, we have the fact that is the best -simultaneous approximation from of .
Let be a sub––algebra of and the restriction of to . Let . We defined the set where is a subset of . We obtain analogously the following theorem.
Theorem 7. Let be a weakly compact subset of and . Then is –simultaneously proximinal in .
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
Research is partially supported by MTM 2012-31286 (Spanish Ministry of Economy and Competitiveness).